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We examine two important measures that can be made in bioarcheology on the remains of human and vertebrate animals. These remains consist of bone, teeth, or hair; each shows growth increments and each can be assayed for isotope ratios and other chemicals in equal intervals along the direction of growth. In each case, the central data is a time series of measurements. The first important measures are spectral estimates in spectral analyses and linear system analyses; we emphasize calculation of periodicities and growth rates as well as the comparison of power in bands. A low frequency band relates to the autonomic nervous system (ANS) control of metabolism and thus provides information about the life history of the individual of archeological interest. Turning to nonlinear system analysis, we discuss the calculation of SM Pinus’ approximate entropy (ApEn) for short or moderate length time series. Like the concept that regular heart R-R interval data may indicate lack of health, low values of ApEn may indicate disrupted metabolism in individuals of archeological interest and even that a tipping point in deteriorating metabolism may have been reached just before death. This adds to the list of causes of death that can be determined from minimal data.

Big data sets are revolutionizing science. They promote insights, facilitate comprehension, and order priorities for further studies using models and powerful computers. In the past decade important advances have been made using big data sets; they range from astronomy to climate change and from biology to geology. Bioarcheology, however, has not benefited from this trend, seemingly, because big data in bioarcheology are difficult to obtain.

Bioarcheology, as defined here, is cross-disciplinary research encompassing the study of human and animal remains. The best preserved tissues are bones, teeth, and occasionally hair.

Here we show that such archived materials provide sufficient data to model life’s activities such as metabolism, growth, and biologic rhythms of individuals who have died decades or even millennia ago.

Many preserved tissues have growth marks left during life which reflect the rates of growth and by extension metabolism. For example, there are “scale like” markings on hair shafts which occur at more or less regular intervals which can be measured (Figure

Human hair with repeat intervals (RIs) marked in green, 50

We hypothesized that the growth lines (GL) in hair, measured by microscopy as a time series, provide direct measurements of hair growth rate, which in turn depends on metabolism and therefore is a proxy for that individual’s metabolism during life [

In death, forensic time series have been linked to ANS function and may reflect on the individual’s life history; these time series include the repeat intervals between growth lines (RIs) in scalp hair expressed as sizes of hair scales measured by microscopy. Also the repeat intervals between

The annual growth rate can often be computed in the time domain.

The forensic time series may be discrete time

Examining the plot

The next step in standardizing the time series

If examination of the plot

The hydrogen isotope ratio measurements (

Fitting the annual sinusoid as well as a trend yields the function of length along the hair in cm: Predicted

This is the growth rate reported in Sharp et al. [

To identified periodicities that are more frequent than annual and less frequent than daily we compute the power spectrum of the discrete time standardized version of our annually adjusted time series using SAS PROC SPECTRA and the Fast Fourier Transform [

To be explicit, let the discrete time, stationary, Gaussian time series representing a series of measured intervals be

Note that each sinusoid

There is a well-known problem with the periodogram as an estimator of the spectral density; it is not consistent; it does not become better as the sample size

The symbol

Let us return to the mammoth example; the estimate of the spectral density of the standardized series in Figure

(a) Hydrogen isotope measured in 0.3 cm intervals of a hair of a Siberian mammoth loaned from the Smithsonian and published in [

There are high frequency

For the low frequency peak, we compute a periodicity of 3.25 weeks. Consider

Similarly, the periodicity of the high frequency peak is 1.2 weeks.

There is a remaining issue; forensic time series are not measured continuously and the use of

To compute the power in a given band of frequencies, the spectral density is integrated over the band; that is, the spectral density times

Thus, the total power or power in frequency bands is obtained as areas under the curve where the units of the

Again the factor of 2 represents the negative frequencies. When the time series is standardized, these formulae are dimensionless and can give a measure of the spectral shape. With these asymptotic means and SEs, one can compute a

The distribution of the AUC estimator is based on the distribution of single estimated spectral densities

In terms of the moments in (

Let us compute the 95% confidence intervals for the low frequency power for the Smithonian mammoth (red line, Figure _{2} = 0.587,

Now the ratio (LF/HF) = AUC_{2}/AUC_{3} = 0.587/0.322 = 1.82. Is this different than 1.0? Here, the high frequency band is 0.27 < _{3} to be (0.20/0.23) _{2}.

For the modified AUC_{3}, we have _{3}) = 0.28 and variance

For an example of comparison between two spectra, we add the data for a hair sample from a Jarkov Siberian mammoth (Figure _{2} = _{2} =

A test of the difference shows no difference:

Longer quiescence in hair growth (telogen [

Let

For the 16th century Spanish royals at the end of life, King Ferrante had an annual hair growth rate of 12 cm/year and Queen Isabella had a growth rate of 2 cm/year. Thus assuming

We now show graphically how the quiescence period

(a) Distance (cm) along a hypothetic hair (blue) that grows continuously for 52 weeks to a length of 20 cm as though there were no quiescence periods; now the quiescence periods are superimposed and marked (red). (b) Incremental weekly growth of the hypothetic hair (blue) with quiescence periods (red), mathematically obtained as the derivative of (a). (c) Observable incremental weekly growth for 39 weeks out of the year, periodic but not a sinusoid though a sinusoid of periodicity 39 weeks fits very well (not shown). (d) Distance (cm) along the observable hair for 39 weeks for an observed length of approximately 16 cm for the year, mathematically obtained as the integral of (c).

However the quiescence periods are not observed; thus the observable result is in Figures

When the periodic function in Figure

We have examined spectral analysis in the frequency domain, which can be considered as linear systems analysis. There are also methods for nonlinear time series analyses and their application to chaos in dynamical systems [

Approximate entropy (ApEn) as described by Pincus et al. [

ApEn depends on three parameters: the length of the time series (

Thus, we use ApEn to measure the logarithmic likelihood that similar patterns of data length (

The “Zweloo woman” was exhumed from a bog in Netherlands in 1951.

We examined six scalp hairs, 2000 years after her death. The approximate entropy (ApEn) was computed for the repeat intervals (RIs) defined by the sizes of hair scales along the length of the hair and for each of her six hairs separately; with

For the methods outlined above, some operational aspects are now considered.

The Nyquist folding frequency probably is not a problem for measured RIs, since generally they are not sampling from a more continuous series. The RIs for hair are deposited in multiple of whole days with the multiple of days being related in an algometric fashion to the species’ body mass; the whole days for periodic deposits to tooth enamel were 1 day for smallest bodied primates to 11 days for largest bodied primates and 8 days for humans [

The data segments caused by missing values are pooled, laid end-to-end. The laying short time series end-to-end (concatenated) to form a longer series may cause difficulty. However, this difficulty is handled automatically in a similar situation for the spectral analysis of heart rate recordings where gaps occur due to technical problems or are introduced to eliminate periods of anomalous heartbeats for separate consideration. We follow this convention unless the difficulties become too large.

An important assumption for the distributions of spectral estimates in the section so named is that the choice of band width for the spectral window is wide enough for that the smoothed estimate of each spectral density function is consistent and narrow enough that estimates of adjacent spectral densities are approximately independent. The series length

Among several additional methods in use for nonlinear time series analysis, there are generalizations of the two basic methods (ApEn and FFT) used herein. First is the replacement of the deterministic rules used in approximate entropy (ApEn) with fuzzy logic rules yielding an improved algorithm (fApEn) that could help with the choice of the tolerance parameter (

For our basic model of quiescence, we assume that the growth rate when the hair is growing is always constant and normal. Then the length of the quiescence intervals is the major effect on the annual growth rate and the effect is algebraic. If the disruption in metabolism affects both

Does quiescence affect the frequency (periodicity) of the low frequency peak, the peak most related to autonomic nervous system (ANS) control? No doubt it does in the same manner that quiescence affects the annual growth cycle. Nonetheless, the computation of the periodicity of the low frequency peak is from the same hair growth record where we compute the growth rate usually reported. As such, it is comparable and useable on its face.

The authors declare that there is no conflict of interests regarding the publication of this paper.